Introduction

Orthonormal wavelet bases have only been available for the last few years, but similar constructions were previously used in mathematics, theoretical physics and signal processing.

It is interesting and surprising that all these related constructions saw the light of day almost simultaneously (in fact, during the 1980s) in constructive field theory, in the geometry of Banach spaces and even in the processing of signals obtained during prospecting trips of the Elf-Aquitaine group (Morlet's work on reflection seismology).

Research workers in the various specialities were hoping to find practical algorithms for decomposing arbitrary functions into sums of special functions which would combine the advantages of the trigonometric and the Haar systems. These systems stand at two extremes, in the following sense: the functions of the trigonometric system are exactly localized by frequency, that is, in the Fourier variable, but have no precise localization in space. On the other hand, the functions of the Haar system (whose definition we recall in Chapter 3) are perfectly localized in space (the x variable) but are badly localized in the Fourier variable (the ξ variable). This is due to two defects of the functions of the Haar system: their lack of regularity and their lack of oscillation.

R. Balian has given the following justification for trying to find Hilbert bases which are simultaneously well-localized in space and in the Fourier variable. “In the theory of communications, it is appropriate to represent an oscillatory signal as the superposition of elementary wavelets possessing both a well-defined frequency and a localization in time.

The purpose of this chapter and the next is to present the background material that will be needed. The topics are standard and a more thorough treatment can be found in many excellent sources, such as Stein [2] and Stein and Weiss [1] for the first half and Hörmander [7, Vol. 1] for the second.

We start out by rapidly going over basic results from real analysis, including standard theorems concerning the Fourier transform in ℝn and Caldéron-Zygmund theory. We then apply this to prove the Hardy-Littlewood-Sobolev inequality. This theorem on fractional integration will be used throughout and we shall also present a simple argument showing how the n-dimensional theorem follows from the original one-dimensional inequality of Hardy and Littlewood. This type of argument will be used again and again. Finally, in the last two sections we give the definition of the wave front set of a distribution and compute the wave front sets of distributions which are given by oscillatory integrals. This will be our first encounter with the cotangent bundle and, as the monograph progresses, this will play an increasingly important role.

We start out with a rapid and somewhat sketchy introduction to Fourier integral operators, emphasizing the role of stationary phase and only presenting material that will be needed later. In Section 2 we give the standard proof of the L2 boundedness of Fourier integral operators whose canonical relations are locally a canonical graph and we state and prove a special case of the composition theorem in which one of the operators is assumed to be of this form. The same proof of course shows that this theorem holds under the weaker assumption that C1 × C2 intersects {(x, ξ, y, η, y, η, z, ζ) : (x, ξ) ∈ T*X\0, (y, η) ∈ T*Y\0, (z, ζ) ∈ T*Z\0} transversally, although it is a little harder to check here that the phase function arising in the proof of the composition theorem is non-degenerate. The next thing we do is to prove the pointwise and LP regularity theorems for Fourier integral operators and show that these are sharp if the operators are conormal with largest possible singular supports. Although this theorem came first, its proof uses the decomposition used in the proof of the maximal theorems for Riesz means and the circular maximal theorem given in Section 2.4. In the last section we apply the estimates for Fourier integral operators to give a proof of Stein's spherical maximal theorem and its variable coefficient generalizations involving the assumption of rotational curvature. In anticipation of the last chapter, we point out how this assumption is inadequate for variable coefficient maximal theorems in the plane.

Except for minor modifications, this monograph represents the lecture notes of a course I gave at UCLA during the winter and spring quarters of 1991. My purpose in the course was to present the necessary background material and to show how ideas from the theory of Fourier integral operators can be useful for studying basic topics in classical analysis, such as oscillatory integrals and maximal functions. The link between the theory of Fourier integral operators and classical analysis is of course not new, since one of the early goals of microlocal analysis was to provide variable coefficient versions of the Fourier transform. However, the primary goal of this subject was to develop tools for the study of partial differential equations and, to some extent, only recently have many classical analysts realized its utility in their subject. In these notes I attempted to stress the unity between these two subjects and only presented the material from microlocal analysis which would be needed for the later applications in Fourier analysis. I did not intend for this course to serve as an introduction to microlocal analysis. For this the reader should be referred to the excellent treatises of Hörmander [5], [7] and Treves [1].

In addition to these sources, I also borrowed heavily from Stein [4]. His work represents lecture notes based on a course which he gave at Princeton while I was his graduate student.

The rest of this course will mainly be concerned with “variable coefficient Fourier analysis”—that is, finding natural variable coefficient versions of the restriction theorem, and so forth. One of our ultimate goals will be to extend these results to the setting of eigenfunction expansions given by the spectral decomposition of a self-adjoint pseudo-differential operator. To state the results, however, and to develop the necessary tools for their study, we need to go over some of the main elements in the theory of pseudo-differential operators. These will be given in Section 1 and our presentation will be a bit sketchy but essentially self-contained. For a more thorough treatment, we refer the reader to the books of Hörmander [7], Taylor [2], and Treves [1]. In Section 2 we present the equivalence of phase function theorem for pseudo-differential operators. This will play an important role in the parametrix construction for the (variable coefficient) half-wave operator. Finally, in Section 3, we present background needed for the study of Fourier analysis on manifolds, such as basic facts about the spectral function. We also present a theorem of Seeley on powers of elliptic differential operators which allows one to reduce questions about the Fourier analysis of higher order elliptic operators to questions about first order operators.

Some Basics

We start out by defining pseudo-differential operators on ℝn.

Many things could be said about the way this book was written but we shall be brief.

It all started with several lecture courses given by N. Varopoulos at Universite Paris VI during the period 1982–87. At the time, Coulhon and SaloffCoste were post-doctoral students and took notes. An early part of these notes appeared for limited circulation in 1986. It was then decided that, when completed, these notes would be published as a set of graduate “Lecture Notes”. The project dragged on for several years; by 1990, through the efforts of Saloff-Coste, enough work had been put into the notes to make them presentable as a real book.

This book is primarily an advanced research monograph. It should be accessible to those graduate students that are prepared to make the personal investment and effort to familiarize themselves with the background material.

N. Varopoulos did very little of the actual writing and did not put any work into the preparation of manuscripts; he is however responsible for most of the new mathematics that is presented here. This mathematical work was done during the 1980s and was built on the following basic material.

Existing semigroup theory, especially Beurling–Deny theory; this is work that was done in the 1950s and 1960s. The work of J. Moser and J. Nash on parabolic equations was also a great inspiration in this context.

The theory of second order subelliptic differential operators and especially the “sum of squares operators”. This is work done in the 1960s by L. Hormander. The Harnack estimates, which are essential for us, were completed by J.-M. Bony a little later. This work has since been further developed by several authors.